Re: Summation problem

• To: mathgroup at smc.vnet.net
• Subject: [mg56675] Re: [mg56621] Summation problem
• Date: Tue, 3 May 2005 05:26:26 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```On Sat, 30 Apr 2005, jaropis wrote:

> Why can't Mathematica sum:
> Sum[I^n/n,{n,1,Infinity}]
> and gives the (incorrect) answer, that this is divergent while it can do:
> Sum[I^(n+1)/n,{n,1,Infinity}]
> correctly?
>
> Jaroslaw Piskorski
>
Hello Jaroslaw Piskorski,

Thank you for reporting this problem with an infinite sum.

This is caused by a failure to detect the conditional convergence
of the sum in your first example, in Mathematica 5.

A workaround for the problem is to replace 'I' by the symbolic
quantity 'x' and then substitute 'x' with 'I', as shown below.

==========================================================
In[1]:= \$Version

Out[1]= 5.1 for Linux (February 20, 2005)

In[2]:= Sum[x^n/n,{n,1,Infinity}]/.{x-> I}

Out[2]= -Log[1 - I]

In[3]:= N[%]

Out[3]= -0.346574 + 0.785398 I

In[4]:= NSum[I^n/n,{n,1,Infinity}]

Out[4]= -0.346574 + 0.785398 I

============================================================

Sorry for the inconvenience caused by this problem.